example of Boolean algebras

Let A be a set. The power setP⁢(A) of A, or the collection of all the subsets of A, together with the operations of union, intersection, and set complement, the empty set∅ and A, is a Boolean algebra. This is the canonical example of a Boolean algebra.

2.

In P⁢(A), let F⁢(A) be the collection of all finite subsets of A, and c⁢F⁢(A) the collection of all cofinite subsets of A. Then F⁢(A)∪c⁢F⁢(A) is a Boolean algebra.

3.

More generally, any field of sets is a Boolean algebra. In particular, any sigma algebra σ in a set is a Boolean algebra.

4.

(product of algebras) Let A and B be Boolean algebras. Then A×B is a Boolean algebra, where

(a,b)∨(c,d)

:=

(a∨c,b∨d),

(1)

(a,b)∧(c,d)

:=

(a∧c,b∧d),

(2)

(a,b)′

:=

(a′,b′).

(3)

5.

More generally, if we have a collection of Boolean algebras Ai, indexed by a set I, then ∏i∈IAi is a Boolean algebra, where the Boolean operations are defined componentwise.

6.

In particular, if A is a Boolean algebra, then set of functions from some non-empty set I to A is also a Boolean algebra, since AI=∏i∈IA.

Let A be a set, and Rn⁢(A) be the set of all n-ary relations on A. Then Rn⁢(A) is a Boolean algebra under the usual set-theoretic operations. The easiest way to see this is to realize that Rn⁢(A)=P⁢(An), the powerset of the n-fold power of A.

Let X be a topological space and A be the collection of all regularly open sets in X. Then A has a Boolean algebraic structure. The meet and the constant operations follow the usual set-theoretic ones: U∧V=U∩V, 0=∅ and 1=X. However, the join ∧ and the complementation′ on A are different. Instead, they are given by